Recently I was reminded of an activity that I used with students that I called The Traffic Jam Activity. View the URLs below if you’re interested in more of the details. There were several ideas that I was working on, but this morning I’m thinking again about concrete (in this case, the little plastic people) manipulatives vs. the virtual (in this case, the Java applet by Mike Morton) manipulatives.

I’m convinced after working for several years with students in a lab environment that using both is better than using one over another. I remember that when I first tried it using both helped me extend the students’ problem solving experience. My students didn’t complain that we were “still” working on a problem or that we were doing a problem “again” if the second time they were trying it with virtual manipulatives. I’m not sure if they considered it to be a different problem or if they liked working on the computer so much they didn’t want to complain or maybe a little of both?

I often observed that for many of my students we had to really work for them to connect the concrete and the virtual experiences. And, in fact, it reminded me of how much we had to work to go from a manipulative environment to a more symbolic environment.

Do you use both kinds of manipulatives when you’re working on a problem? Do you have students have concrete Activity Pattern Blocks (for example) as your students have that applet in front of them? or Tangrams? or Algebra Tiles? or Dice?

I wonder if the conversations that you might have with students as they make connections between the concrete and the virtual might be a starting point for the conversations that you might have with them about going from the concrete to the abstract?

What are your thoughts?

**Some “Traffic Jam” links** in case you are interested:

- ON-Math Fall 2002, Volume 1, Number 1: Developing Algebraic Thinking [PDF version]
- Post-lesson Interview with questions posed to me by Ihor Charischak
- Suzanne’s Lessons: Traffic Jam Activity
- Technology Problems of the Week (tPoWs): Traffic Jam

Teaching solving equations for a variable is a challenge in 6th grade. Students are at a variety of thinking levels from concrete to the abstract. I teach with concrete manipulatives and drawing balance scales on the board to match equations. The biggest “aha” leap my students have made with learning about solving for a variable has been with NLVM Algebra Balance Scales.

http://nlvm.usu.edu/en/nav/frames_asid_201_g_3_t_2.html?open=instructions&from=category_g_3_t_2.html

This is a virtual manipulative I use on the smartboard. It seems to be concrete (students can touch and drag), symbolic (equation is what is being manipulated) and abstract in understanding what is happening. My students want to do this endlessly. I think because they can see the connection between the concrete and the abstract.

We have discussions about how the virtual manipulative represents what we are doing mathematically on paper as we solve each equation. Students try to solve the equation first on paper and then take turns solving it step by step on the smartboard. It is a simple but powerful manipulative. Our discussions help those less abstract thinkers reach what we are doing. Interestingly, the struggling students want to try and are not afraid to make a mistake on the smartboard. I think because they can experiment and “undo” when they see the balance scale is going the wrong way. At the same time the class is disucssing what is working and not working and why. It is powerful. Thanks for reminding me to start my year with this fun activity! ;)